Description: The complement of an open set is closed. (Contributed by NM, 6-Oct-2006)
Ref | Expression | ||
---|---|---|---|
Hypothesis | iscld.1 | ⊢ 𝑋 = ∪ 𝐽 | |
Assertion | opncld | ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ) → ( 𝑋 ∖ 𝑆 ) ∈ ( Clsd ‘ 𝐽 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscld.1 | ⊢ 𝑋 = ∪ 𝐽 | |
2 | simpr | ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ) → 𝑆 ∈ 𝐽 ) | |
3 | 1 | eltopss | ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ) → 𝑆 ⊆ 𝑋 ) |
4 | 1 | isopn2 | ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑆 ∈ 𝐽 ↔ ( 𝑋 ∖ 𝑆 ) ∈ ( Clsd ‘ 𝐽 ) ) ) |
5 | 3 4 | syldan | ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ) → ( 𝑆 ∈ 𝐽 ↔ ( 𝑋 ∖ 𝑆 ) ∈ ( Clsd ‘ 𝐽 ) ) ) |
6 | 2 5 | mpbid | ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ) → ( 𝑋 ∖ 𝑆 ) ∈ ( Clsd ‘ 𝐽 ) ) |